The constant function f(x) = 0 will give zero no matter what function
g(x) it is integrated with. Does this mean that the constant function
zero is orthogonal to all functions? Also, what could be the geometrical
interpretation of orthogonal functions?

In calculus, we were looking for the solutions for a third degree
polynomial equation. Using my TI-85 to find the solutions, I stumbled
upon an interesting observation. Given a cubic of the form ax^3 + bx + c,
the absolute value of the sum of any two zeros seems to be equal to the
absolute value of the third. However, I have so far been unable to give a
general proof.

When calculating derivatives by hand using the definition given in
first principles, we seem to eventually just let h = 0. But doesn't
that violate the rule about division by 0 since the definition starts
with a denominator of h?

I am to find the derivative of the function (x-6)(x+1)/(x-6) at x=6. I
simplified the function to a linear function x+1 with a "hole" at x=6.
Then I tried to take the derivative.... but my calculator says division
by zero, infinite result...